Find the true discount, Banker’s discount and Banker’s gain on a bill of Rs.4,240 due 6 months hence at 9% p.a.

Answer

To solve the problem, we need to find the True Discount, Banker's Discount, and Banker's Gain on a bill of Rs. 4,240 due in 6 months at an interest rate of 9% per annum.

### Step 1: Calculate the Banker's Discount (BD)

The formula for calculating the Banker's Discount is:
\[ \text{Banker's Discount} = \text{Principal} \times \text{Rate} \times \text{Time} \]
Where:
- Principal (P) = Rs. 4,240
- Rate (R) = 9% per annum
- Time (T) = 6 months = 0.5 years

Substituting the values into the formula:
\[ \text{Banker's Discount} = 4240 \times \frac{9}{100} \times 0.5 \]
\[ = 4240 \times 0.09 \times 0.5 \]
\[ = 4240 \times 0.045 \]
\[ = 190.8 \]

### Step 2: Calculate the True Discount (TD)

The relationship between True Discount (TD) and Banker's Discount (BD) is given by:
\[ \text{Banker's Discount} = \text{True Discount} + \text{Simple Interest on True Discount} \]
Let True Discount be \( x \). The Simple Interest on True Discount can be calculated as:
\[ \text{Simple Interest} = x \times \frac{R}{100} \times T \]
Substituting the values:
\[ \text{Simple Interest} = x \times \frac{9}{100} \times 0.5 = \frac{9x}{200} \]

Now substituting into the equation:
\[ 190.8 = x + \frac{9x}{200} \]
To eliminate the fraction, multiply through by 200:
\[ 190.8 \times 200 = 200x + 9x \]
\[ 38160 = 209x \]
Now, solve for \( x \):
\[ x = \frac{38160}{209} \approx 182.58 \]

### Step 3: Calculate the Banker's Gain (BG)

Banker's Gain is calculated as:
\[ \text{Banker's Gain} = \text{Banker's Discount} - \text{True Discount} \]
Substituting the values we found:
\[ \text{Banker's Gain} = 190.8 - 182.58 = 8.22 \]

### Final Results:
- Banker's Discount = Rs. 190.8
- True Discount = Rs. 182.58
- Banker's Gain = Rs. 8.22